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450Head Start in an Exponential RaceLet X \sim Exp ( ) and Y \sim Exp ( ) be independent. Player A finishes at time X and player B finishes at time Y + c where c > 0 is a head start for player A (player B starts c time units later). (a) Derive P(X < Y + c) — the probability that A finishes before B. (b) Show that as c 0, the result recovers the standard competing-exponentials formula. (c) Evaluate for = 3, = 2, c = 1 and interpret how the head start affects A's winning probability.概率困难multi part未尝试面试订阅6008Which of Three Feeds Ticks FirstThree independent market-data feeds emit ticks as Poisson processes with rates 5, 8, and 2 ticks per second. What is the probability that the very next tick across all feeds comes from the rate-8 feed?概率简单derivation未尝试面试订阅6012A Three-Arrival Head StartAggressive and passive child orders fill as independent Poisson processes with rates \lambda A=10 and \lambda B=5 per minute. What is the probability that the first three fills in the merged stream are all aggressive (stream A)?概率简单derivation未尝试面试订阅